Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs

Augmented Lagrangian dual for nonconvex minimax fractional programs and proximal bundle algorithms for its resolution

“…In Section 4, we introduce the general approximating proximal augmented Lagrangian methods for solving the augmented Lagrangian dual of GFP. Their convergence proofs are analogous to those used in our work [12]. Section 5 is devoted to effective constructions of approximating functions of the dual objective function.…”

“…In the literature, there have been two types of algorithms for solving a GFP, primal Dinkelbach-type algorithms [10,17,18,19,21,40,39,51,28,13]; and dual algorithms [2,3,7,8,9,11,12,14,15,16,20,22,23,31]. See also [45,46,50,49,47,48] for more references on fractional programming.…”

“…In [37], Rockafellar pointed out that usually a duality gap occurs if one tackles the problem with its ordinary Lagrangian function, and introduced an augmented Lagrangian with quadratic terms to remove this gap. In this direction, this paper aims to complete earlier works [14,11,12,15] dealing with Lagrangian duality for GFP by defining an augmented Lagrangian dual for the latter, and by developing similar algorithms for its resolution. To construct this dual we use the theory developed by Rockafellar in [37].…”

“…Our approach is then to apply the augmented Lagrangian theory of [37] to the equivalent form (Pλ) of (P ), to obtain an augmented Lagrangian dual for GFP. This is the way we followed in [14,11,12,15] to develop Lagrangian duality and algorithms for GFP.…”

“…In the same spirit, another primal bundle method based on the extended method of centers [40], was proposed in [2] to deal with nonlinearly constrained GFP. On the other hand, dual bundle algorithms have been proposed in [11,12] to, respectively, solve the partial dual problem [7,53] and the standard dual problem [8]. The proposed bundle algorithm generates two sequences, one converges to the optimal value of (P ) and the other converges to an optimal solution of the augmented Lagrangian dual problem.…”